LMFDB - Embedded newform 2475.1.fy.b.428.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2475,1,Mod(263,2475)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2475, base_ring=CyclotomicField(60))

chi = DirichletCharacter(H, H._module([10, 57, 30]))

N = Newforms(chi, 1, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2475.263");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2475 = 3^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2475.fy (of order $60$, degree $16$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$1.23518590627$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\Q(\zeta_{60})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + x^{14} - x^{10} - x^{8} - x^{6} + x^{2} + 1 $ x^16 + x^14 - x^10 - x^8 - x^6 + x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{60}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{60} + \cdots)$

Embedding invariants

Embedding label			428.1
Root			$0.207912 + 0.978148i$ of defining polynomial
Character	$\chi$	$=$	2475.428
Dual form			2475.1.fy.b.2342.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.743145 - 0.669131i) q^{3} +(0.406737 + 0.913545i) q^{4} -1.00000i q^{5} +(0.104528 - 0.994522i) q^{9} +O(q^{10})$	\(q+(0.743145 - 0.669131i) q^{3} +(0.406737 + 0.913545i) q^{4} -1.00000i q^{5} +(0.104528 - 0.994522i) q^{9} +(-0.207912 - 0.978148i) q^{11} +(0.913545 + 0.406737i) q^{12} +(-0.669131 - 0.743145i) q^{15} +(-0.669131 + 0.743145i) q^{16} +(0.913545 - 0.406737i) q^{20} +(1.97267 + 0.103383i) q^{23} -1.00000 q^{25} +(-0.587785 - 0.809017i) q^{27} +(0.0434654 + 0.413545i) q^{31} +(-0.809017 - 0.587785i) q^{33} +(0.951057 - 0.309017i) q^{36} +(-1.72129 - 0.877042i) q^{37} +(0.809017 - 0.587785i) q^{44} +(-0.994522 - 0.104528i) q^{45} +(0.402280 - 0.325760i) q^{47} +1.00000i q^{48} +(0.866025 + 0.500000i) q^{49} +(1.07587 - 0.170401i) q^{53} +(-0.978148 + 0.207912i) q^{55} +(-1.30902 - 0.278240i) q^{59} +(0.406737 - 0.913545i) q^{60} +(-0.951057 - 0.309017i) q^{64} +(0.557008 + 0.451057i) q^{67} +(1.53516 - 1.24314i) q^{69} +(-0.873619 + 1.20243i) q^{71} +(-0.743145 + 0.669131i) q^{75} +(0.743145 + 0.669131i) q^{80} +(-0.978148 - 0.207912i) q^{81} +(-0.363271 + 1.11803i) q^{89} +(0.707912 + 1.84417i) q^{92} +(0.309017 + 0.278240i) q^{93} +(1.05558 + 1.30354i) q^{97} +(-0.994522 + 0.104528i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 2 q^{9}+O(q^{10}) $ 16 * q - 2 * q^9	$ 16 q - 2 q^{9} + 2 q^{12} - 2 q^{15} - 2 q^{16} + 2 q^{20} - 2 q^{23} - 16 q^{25} - 4 q^{33} + 2 q^{37} + 4 q^{44} + 4 q^{47} + 2 q^{53} + 2 q^{55} - 12 q^{59} + 4 q^{67} + 2 q^{69} + 2 q^{81} + 8 q^{92} - 4 q^{93} - 2 q^{97}+O(q^{100}) $ 16 * q - 2 * q^9 + 2 * q^12 - 2 * q^15 - 2 * q^16 + 2 * q^20 - 2 * q^23 - 16 * q^25 - 4 * q^33 + 2 * q^37 + 4 * q^44 + 4 * q^47 + 2 * q^53 + 2 * q^55 - 12 * q^59 + 4 * q^67 + 2 * q^69 + 2 * q^81 + 8 * q^92 - 4 * q^93 - 2 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2475\mathbb{Z}\right)^\times$.

$n$	$551$	$2026$	$2377$
$\chi(n)$	$e\left(\frac{5}{6}\right)$	$-1$	$e\left(\frac{7}{20}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0		−0.838671	−	0.544639i	$-0.816667\pi$
$2$		0			0		0.838671	+	0.544639i	$0.183333\pi$
$3$	0.743145	−	0.669131i	0.743145	−	0.669131i
$3$	0.743145	−	0.669131i	0.743145	−	0.669131i
$4$	0.406737	+	0.913545i	0.406737	+	0.913545i
$5$		−	1.00000i		−	1.00000i
$5$		−	1.00000i		−	1.00000i
$6$		0			0
$7$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$7$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$8$		0			0
$9$	0.104528	−	0.994522i	0.104528	−	0.994522i
$10$		0			0
$11$	−0.207912	−	0.978148i	−0.207912	−	0.978148i
$11$	−0.207912	−	0.978148i	−0.207912	−	0.978148i
$12$	0.913545	+	0.406737i	0.913545	+	0.406737i
$13$		0			0		−0.544639	−	0.838671i	$-0.683333\pi$
$13$		0			0		0.544639	+	0.838671i	$0.316667\pi$
$14$		0			0
$15$	−0.669131	−	0.743145i	−0.669131	−	0.743145i
$16$	−0.669131	+	0.743145i	−0.669131	+	0.743145i
$17$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$17$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$18$		0			0
$19$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$19$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$20$	0.913545	−	0.406737i	0.913545	−	0.406737i
$21$		0			0
$22$		0			0
$23$	1.97267	+	0.103383i	1.97267	+	0.103383i	0.994522	−	0.104528i	$-0.0333333\pi$
$23$	1.97267	+	0.103383i	1.97267	+	0.103383i	0.978148	+	0.207912i	$0.0666667\pi$
$24$		0			0
$25$	−1.00000			−1.00000
$26$		0			0
$27$	−0.587785	−	0.809017i	−0.587785	−	0.809017i
$28$		0			0
$29$		0			0		0.104528	−	0.994522i	$-0.466667\pi$
$29$		0			0		−0.104528	+	0.994522i	$0.533333\pi$
$30$		0			0
$31$	0.0434654	+	0.413545i	0.0434654	+	0.413545i	0.994522	+	0.104528i	$0.0333333\pi$
$31$	0.0434654	+	0.413545i	0.0434654	+	0.413545i	−0.951057	+	0.309017i	$0.900000\pi$
$32$		0			0
$33$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$34$		0			0
$35$		0			0
$36$	0.951057	−	0.309017i	0.951057	−	0.309017i
$37$	−1.72129	−	0.877042i	−1.72129	−	0.877042i	−0.978148	−	0.207912i	$-0.933333\pi$
$37$	−1.72129	−	0.877042i	−1.72129	−	0.877042i	−0.743145	−	0.669131i	$-0.766667\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		0.207912	−	0.978148i	$-0.433333\pi$
$41$		0			0		−0.207912	+	0.978148i	$0.566667\pi$
$42$		0			0
$43$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$43$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$44$	0.809017	−	0.587785i	0.809017	−	0.587785i
$45$	−0.994522	−	0.104528i	−0.994522	−	0.104528i
$46$		0			0
$47$	0.402280	−	0.325760i	0.402280	−	0.325760i	−0.406737	−	0.913545i	$-0.633333\pi$
$47$	0.402280	−	0.325760i	0.402280	−	0.325760i	0.809017	+	0.587785i	$0.200000\pi$
$48$			1.00000i			1.00000i
$49$	0.866025	+	0.500000i	0.866025	+	0.500000i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	1.07587	−	0.170401i	1.07587	−	0.170401i	0.406737	−	0.913545i	$-0.366667\pi$
$53$	1.07587	−	0.170401i	1.07587	−	0.170401i	0.669131	+	0.743145i	$0.266667\pi$
$54$		0			0
$55$	−0.978148	+	0.207912i	−0.978148	+	0.207912i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−1.30902	−	0.278240i	−1.30902	−	0.278240i	−0.500000	−	0.866025i	$-0.666667\pi$
$59$	−1.30902	−	0.278240i	−1.30902	−	0.278240i	−0.809017	+	0.587785i	$0.800000\pi$
$60$	0.406737	−	0.913545i	0.406737	−	0.913545i
$61$		0			0		0.978148	−	0.207912i	$-0.0666667\pi$
$61$		0			0		−0.978148	+	0.207912i	$0.933333\pi$
$62$		0			0
$63$		0			0
$64$	−0.951057	−	0.309017i	−0.951057	−	0.309017i
$65$		0			0
$66$		0			0
$67$	0.557008	+	0.451057i	0.557008	+	0.451057i	0.866025	−	0.500000i	$-0.166667\pi$
$67$	0.557008	+	0.451057i	0.557008	+	0.451057i	−0.309017	+	0.951057i	$0.600000\pi$
$68$		0			0
$69$	1.53516	−	1.24314i	1.53516	−	1.24314i
$70$		0			0
$71$	−0.873619	+	1.20243i	−0.873619	+	1.20243i	0.104528	+	0.994522i	$0.466667\pi$
$71$	−0.873619	+	1.20243i	−0.873619	+	1.20243i	−0.978148	+	0.207912i	$0.933333\pi$
$72$		0			0
$73$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$73$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$74$		0			0
$75$	−0.743145	+	0.669131i	−0.743145	+	0.669131i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		−0.994522	−	0.104528i	$-0.966667\pi$
$79$		0			0		0.994522	+	0.104528i	$0.0333333\pi$
$80$	0.743145	+	0.669131i	0.743145	+	0.669131i
$81$	−0.978148	−	0.207912i	−0.978148	−	0.207912i
$82$		0			0
$83$		0			0		0.933580	−	0.358368i	$-0.116667\pi$
$83$		0			0		−0.933580	+	0.358368i	$0.883333\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−0.363271	+	1.11803i	−0.363271	+	1.11803i	0.587785	+	0.809017i	$0.300000\pi$
$89$	−0.363271	+	1.11803i	−0.363271	+	1.11803i	−0.951057	+	0.309017i	$0.900000\pi$
$90$		0			0
$91$		0			0
$92$	0.707912	+	1.84417i	0.707912	+	1.84417i
$93$	0.309017	+	0.278240i	0.309017	+	0.278240i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	1.05558	+	1.30354i	1.05558	+	1.30354i	0.951057	+	0.309017i	$0.100000\pi$
$97$	1.05558	+	1.30354i	1.05558	+	1.30354i	0.104528	+	0.994522i	$0.466667\pi$
$98$		0			0
$99$	−0.994522	+	0.104528i	−0.994522	+	0.104528i
$100$	−0.406737	−	0.913545i	−0.406737	−	0.913545i
$101$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$101$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$102$		0			0
$103$	1.45106	+	0.557008i	1.45106	+	0.557008i	0.951057	−	0.309017i	$-0.100000\pi$
$103$	1.45106	+	0.557008i	1.45106	+	0.557008i	0.500000	+	0.866025i	$0.333333\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$107$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$108$	0.500000	−	0.866025i	0.500000	−	0.866025i
$109$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$109$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$110$		0			0
$111$	−1.86603	+	0.500000i	−1.86603	+	0.500000i
$112$		0			0
$113$	1.56593	−	1.01693i	1.56593	−	1.01693i	0.587785	−	0.809017i	$-0.300000\pi$
$113$	1.56593	−	1.01693i	1.56593	−	1.01693i	0.978148	−	0.207912i	$-0.0666667\pi$
$114$		0			0
$115$	0.103383	−	1.97267i	0.103383	−	1.97267i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−0.913545	+	0.406737i	−0.913545	+	0.406737i
$122$		0			0
$123$		0			0
$124$	−0.360114	+	0.207912i	−0.360114	+	0.207912i
$125$			1.00000i			1.00000i
$126$		0			0
$127$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$127$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		0.994522	−	0.104528i	$-0.0333333\pi$
$131$		0			0		−0.994522	+	0.104528i	$0.966667\pi$
$132$	0.207912	−	0.978148i	0.207912	−	0.978148i
$133$		0			0
$134$		0			0
$135$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$136$		0			0
$137$	−1.55216	+	0.0813454i	−1.55216	+	0.0813454i	−0.809017	−	0.587785i	$-0.800000\pi$
$137$	−1.55216	+	0.0813454i	−1.55216	+	0.0813454i	−0.743145	+	0.669131i	$0.766667\pi$
$138$		0			0
$139$		0			0		0.743145	−	0.669131i	$-0.233333\pi$
$139$		0			0		−0.743145	+	0.669131i	$0.766667\pi$
$140$		0			0
$141$	0.0809764	−	0.511265i	0.0809764	−	0.511265i
$142$		0			0
$143$		0			0
$144$	0.669131	+	0.743145i	0.669131	+	0.743145i
$145$		0			0
$146$		0			0
$147$	0.978148	−	0.207912i	0.978148	−	0.207912i
$148$	0.101105	−	1.92920i	0.101105	−	1.92920i
$149$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$149$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$150$		0			0
$151$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$151$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	0.413545	−	0.0434654i	0.413545	−	0.0434654i
$156$		0			0
$157$	0.483257	+	1.80354i	0.483257	+	1.80354i	0.587785	+	0.809017i	$0.300000\pi$
$157$	0.483257	+	1.80354i	0.483257	+	1.80354i	−0.104528	+	0.994522i	$0.533333\pi$
$158$		0			0
$159$	0.685505	−	0.846528i	0.685505	−	0.846528i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	0.0475201	−	0.0932634i	0.0475201	−	0.0932634i	−0.866025	−	0.500000i	$-0.833333\pi$
$163$	0.0475201	−	0.0932634i	0.0475201	−	0.0932634i	0.913545	+	0.406737i	$0.133333\pi$
$164$		0			0
$165$	−0.587785	+	0.809017i	−0.587785	+	0.809017i
$166$		0			0
$167$		0			0		0.629320	−	0.777146i	$-0.283333\pi$
$167$		0			0		−0.629320	+	0.777146i	$0.716667\pi$
$168$		0			0
$169$	−0.406737	+	0.913545i	−0.406737	+	0.913545i
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0		0.544639	−	0.838671i	$-0.316667\pi$
$173$		0			0		−0.544639	+	0.838671i	$0.683333\pi$
$174$		0			0
$175$		0			0
$176$	0.866025	+	0.500000i	0.866025	+	0.500000i
$177$	−1.15897	+	0.669131i	−1.15897	+	0.669131i
$178$		0			0
$179$	−1.60917	−	1.16913i	−1.60917	−	1.16913i	−0.866025	−	0.500000i	$-0.833333\pi$
$179$	−1.60917	−	1.16913i	−1.60917	−	1.16913i	−0.743145	−	0.669131i	$-0.766667\pi$
$180$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$181$	−1.40126	+	1.01807i	−1.40126	+	1.01807i	−0.406737	+	0.913545i	$0.633333\pi$
$181$	−1.40126	+	1.01807i	−1.40126	+	1.01807i	−0.994522	+	0.104528i	$0.966667\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−0.877042	+	1.72129i	−0.877042	+	1.72129i
$186$		0			0
$187$		0			0
$188$	0.461219	+	0.235003i	0.461219	+	0.235003i
$189$		0			0
$190$		0			0
$191$	0.155360	+	0.139886i	0.155360	+	0.139886i	0.743145	−	0.669131i	$-0.233333\pi$
$191$	0.155360	+	0.139886i	0.155360	+	0.139886i	−0.587785	+	0.809017i	$0.700000\pi$
$192$	−0.913545	+	0.406737i	−0.913545	+	0.406737i
$193$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$193$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$194$		0			0
$195$		0			0
$196$	−0.104528	+	0.994522i	−0.104528	+	0.994522i
$197$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$197$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$198$		0			0
$199$		−	0.813473i		−	0.813473i	−0.913545	−	0.406737i	$-0.866667\pi$
$199$		−	0.813473i		−	0.813473i	0.913545	−	0.406737i	$-0.133333\pi$
$200$		0			0
$201$	0.715754	−	0.0375111i	0.715754	−	0.0375111i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	0.309017	−	1.95106i	0.309017	−	1.95106i
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		−0.669131	−	0.743145i	$-0.733333\pi$
$211$		0			0		0.669131	+	0.743145i	$0.266667\pi$
$212$	0.593263	+	0.913545i	0.593263	+	0.913545i
$213$	0.155360	+	1.47815i	0.155360	+	1.47815i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$	−0.587785	−	0.809017i	−0.587785	−	0.809017i
$221$		0			0
$222$		0			0
$223$	−1.49452	−	0.970554i	−1.49452	−	0.970554i	−0.994522	−	0.104528i	$-0.966667\pi$
$223$	−1.49452	−	0.970554i	−1.49452	−	0.970554i	−0.500000	−	0.866025i	$-0.666667\pi$
$224$		0			0
$225$	−0.104528	+	0.994522i	−0.104528	+	0.994522i
$226$		0			0
$227$		0			0		−0.838671	−	0.544639i	$-0.816667\pi$
$227$		0			0		0.838671	+	0.544639i	$0.183333\pi$
$228$		0			0
$229$	0.478148	+	1.07394i	0.478148	+	1.07394i	0.978148	+	0.207912i	$0.0666667\pi$
$229$	0.478148	+	1.07394i	0.478148	+	1.07394i	−0.500000	+	0.866025i	$0.666667\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$233$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$234$		0			0
$235$	−0.325760	−	0.402280i	−0.325760	−	0.402280i
$236$	−0.278240	−	1.30902i	−0.278240	−	1.30902i
$237$		0			0
$238$		0			0
$239$		0			0		−0.669131	−	0.743145i	$-0.733333\pi$
$239$		0			0		0.669131	+	0.743145i	$0.266667\pi$
$240$	1.00000			1.00000
$241$		0			0		0.669131	−	0.743145i	$-0.266667\pi$
$241$		0			0		−0.669131	+	0.743145i	$0.733333\pi$
$242$		0			0
$243$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$244$		0			0
$245$	0.500000	−	0.866025i	0.500000	−	0.866025i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$			1.90211i			1.90211i	0.309017	+	0.951057i	$0.400000\pi$
$251$			1.90211i			1.90211i	−0.309017	+	0.951057i	$0.600000\pi$
$252$		0			0
$253$	−0.309017	−	1.95106i	−0.309017	−	1.95106i
$254$		0			0
$255$		0			0
$256$	−0.104528	−	0.994522i	−0.104528	−	0.994522i
$257$	0.302208	−	0.0809764i	0.302208	−	0.0809764i	−0.104528	−	0.994522i	$-0.533333\pi$
$257$	0.302208	−	0.0809764i	0.302208	−	0.0809764i	0.406737	+	0.913545i	$0.366667\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.0523360	−	0.998630i	$-0.516667\pi$
$263$		0			0		0.0523360	+	0.998630i	$0.483333\pi$
$264$		0			0
$265$	−0.170401	−	1.07587i	−0.170401	−	1.07587i
$266$		0			0
$267$	0.478148	+	1.07394i	0.478148	+	1.07394i
$268$	−0.185505	+	0.692314i	−0.185505	+	0.692314i
$269$	1.47815	−	1.07394i	1.47815	−	1.07394i	0.500000	−	0.866025i	$-0.333333\pi$
$269$	1.47815	−	1.07394i	1.47815	−	1.07394i	0.978148	−	0.207912i	$-0.0666667\pi$
$270$		0			0
$271$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$271$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	0.207912	+	0.978148i	0.207912	+	0.978148i
$276$	1.76007	+	0.896802i	1.76007	+	0.896802i
$277$		0			0		0.544639	−	0.838671i	$-0.316667\pi$
$277$		0			0		−0.544639	+	0.838671i	$0.683333\pi$
$278$		0			0
$279$	0.415823			0.415823
$280$		0			0
$281$		0			0		0.406737	−	0.913545i	$-0.366667\pi$
$281$		0			0		−0.406737	+	0.913545i	$0.633333\pi$
$282$		0			0
$283$		0			0		0.629320	−	0.777146i	$-0.283333\pi$
$283$		0			0		−0.629320	+	0.777146i	$0.716667\pi$
$284$	−1.45381	−	0.309017i	−1.45381	−	0.309017i
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	0.951057	+	0.309017i	0.951057	+	0.309017i
$290$		0			0
$291$	1.65669	+	0.262394i	1.65669	+	0.262394i
$292$		0			0
$293$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$293$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$294$		0			0
$295$	−0.278240	+	1.30902i	−0.278240	+	1.30902i
$296$		0			0
$297$	−0.669131	+	0.743145i	−0.669131	+	0.743145i
$298$		0			0
$299$		0			0
$300$	−0.913545	−	0.406737i	−0.913545	−	0.406737i
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$307$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$308$		0			0
$309$	1.45106	−	0.557008i	1.45106	−	0.557008i
$310$		0			0
$311$	0.604528	−	0.544320i	0.604528	−	0.544320i	−0.309017	−	0.951057i	$-0.600000\pi$
$311$	0.604528	−	0.544320i	0.604528	−	0.544320i	0.913545	+	0.406737i	$0.133333\pi$
$312$		0			0
$313$	0.312440	−	0.0163743i	0.312440	−	0.0163743i	0.104528	−	0.994522i	$-0.466667\pi$
$313$	0.312440	−	0.0163743i	0.312440	−	0.0163743i	0.207912	+	0.978148i	$0.433333\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−0.325391	−	0.847673i	−0.325391	−	0.847673i	−0.994522	−	0.104528i	$-0.966667\pi$
$317$	−0.325391	−	0.847673i	−0.325391	−	0.847673i	0.669131	−	0.743145i	$-0.266667\pi$
$318$		0			0
$319$		0			0
$320$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−0.207912	−	0.978148i	−0.207912	−	0.978148i
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−1.81708	−	0.809017i	−1.81708	−	0.809017i	−0.951057	−	0.309017i	$-0.900000\pi$
$331$	−1.81708	−	0.809017i	−1.81708	−	0.809017i	−0.866025	−	0.500000i	$-0.833333\pi$
$332$		0			0
$333$	−1.05216	+	1.62019i	−1.05216	+	1.62019i
$334$		0			0
$335$	0.451057	−	0.557008i	0.451057	−	0.557008i
$336$		0			0
$337$		0			0		0.838671	−	0.544639i	$-0.183333\pi$
$337$		0			0		−0.838671	+	0.544639i	$0.816667\pi$
$338$		0			0
$339$	0.483257	−	1.80354i	0.483257	−	1.80354i
$340$		0			0
$341$	0.395472	−	0.128496i	0.395472	−	0.128496i
$342$		0			0
$343$		0			0
$344$		0			0
$345$	−1.24314	−	1.53516i	−1.24314	−	1.53516i
$346$		0			0
$347$		0			0		−0.933580	−	0.358368i	$-0.883333\pi$
$347$		0			0		0.933580	+	0.358368i	$0.116667\pi$
$348$		0			0
$349$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$349$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	0.889993	+	1.09905i	0.889993	+	1.09905i	0.994522	+	0.104528i	$0.0333333\pi$
$353$	0.889993	+	1.09905i	0.889993	+	1.09905i	−0.104528	+	0.994522i	$0.533333\pi$
$354$		0			0
$355$	1.20243	+	0.873619i	1.20243	+	0.873619i
$356$	−1.16913	+	0.122881i	−1.16913	+	0.122881i
$357$		0			0
$358$		0			0
$359$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$359$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$360$		0			0
$361$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$362$		0			0
$363$	−0.406737	+	0.913545i	−0.406737	+	0.913545i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	1.17504	−	0.451057i	1.17504	−	0.451057i	0.309017	−	0.951057i	$-0.400000\pi$
$367$	1.17504	−	0.451057i	1.17504	−	0.451057i	0.866025	+	0.500000i	$0.166667\pi$
$368$	−1.39680	+	1.39680i	−1.39680	+	1.39680i
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−0.128496	+	0.395472i	−0.128496	+	0.395472i
$373$		0			0		0.0523360	−	0.998630i	$-0.483333\pi$
$373$		0			0		−0.0523360	+	0.998630i	$0.516667\pi$
$374$		0			0
$375$	0.669131	+	0.743145i	0.669131	+	0.743145i
$376$		0			0
$377$		0			0
$378$		0			0
$379$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$379$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	1.55216	+	1.25692i	1.55216	+	1.25692i	0.809017	+	0.587785i	$0.200000\pi$
$383$	1.55216	+	1.25692i	1.55216	+	1.25692i	0.743145	+	0.669131i	$0.233333\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$	−0.761497	+	1.49452i	−0.761497	+	1.49452i
$389$	−1.91355	+	0.406737i	−1.91355	+	0.406737i	−0.913545	+	0.406737i	$0.866667\pi$
$389$	−1.91355	+	0.406737i	−1.91355	+	0.406737i	−1.00000			$\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$397$	1.24314	−	0.196895i	1.24314	−	0.196895i	0.500000	−	0.866025i	$-0.333333\pi$
$397$	1.24314	−	0.196895i	1.24314	−	0.196895i	0.743145	+	0.669131i	$0.233333\pi$
$398$		0			0
$399$		0			0
$400$	0.669131	−	0.743145i	0.669131	−	0.743145i
$401$	−1.69420	−	0.978148i	−1.69420	−	0.978148i	−0.951057	−	0.309017i	$-0.900000\pi$
$401$	−1.69420	−	0.978148i	−1.69420	−	0.978148i	−0.743145	−	0.669131i	$-0.766667\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	−0.207912	+	0.978148i	−0.207912	+	0.978148i
$406$		0			0
$407$	−0.500000	+	1.86603i	−0.500000	+	1.86603i
$408$		0			0
$409$		0			0		0.207912	−	0.978148i	$-0.433333\pi$
$409$		0			0		−0.207912	+	0.978148i	$0.566667\pi$
$410$		0			0
$411$	−1.09905	+	1.09905i	−1.09905	+	1.09905i
$412$	0.0813454	+	1.55216i	0.0813454	+	1.55216i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−0.0434654	−	0.413545i	−0.0434654	−	0.413545i	−0.994522	−	0.104528i	$-0.966667\pi$
$419$	−0.0434654	−	0.413545i	−0.0434654	−	0.413545i	0.951057	−	0.309017i	$-0.100000\pi$
$420$		0			0
$421$	−0.207912	+	1.97815i	−0.207912	+	1.97815i			1.00000i	$0.5\pi$
$421$	−0.207912	+	1.97815i	−0.207912	+	1.97815i	−0.207912	+	0.978148i	$0.566667\pi$
$422$		0			0
$423$	−0.281926	−	0.434128i	−0.281926	−	0.434128i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$431$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$432$	0.994522	+	0.104528i	0.994522	+	0.104528i
$433$	−1.39680	−	0.221232i	−1.39680	−	0.221232i	−0.587785	−	0.809017i	$-0.700000\pi$
$433$	−1.39680	−	0.221232i	−1.39680	−	0.221232i	−0.809017	+	0.587785i	$0.800000\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		−0.207912	−	0.978148i	$-0.566667\pi$
$439$		0			0		0.207912	+	0.978148i	$0.433333\pi$
$440$		0			0
$441$	0.587785	−	0.809017i	0.587785	−	0.809017i
$442$		0			0
$443$	−1.92920	−	0.516929i	−1.92920	−	0.516929i	−0.978148	−	0.207912i	$-0.933333\pi$
$443$	−1.92920	−	0.516929i	−1.92920	−	0.516929i	−0.951057	−	0.309017i	$-0.900000\pi$
$444$	−1.21575	−	1.50133i	−1.21575	−	1.50133i
$445$	1.11803	+	0.363271i	1.11803	+	0.363271i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−0.813473			−0.813473			−0.406737	−	0.913545i	$-0.633333\pi$
$449$	−0.813473			−0.813473			−0.406737	+	0.913545i	$0.633333\pi$
$450$		0			0
$451$		0			0
$452$	1.56593	+	1.01693i	1.56593	+	1.01693i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$457$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$458$		0			0
$459$		0			0
$460$	1.84417	−	0.707912i	1.84417	−	0.707912i
$461$		0			0		−0.207912	−	0.978148i	$-0.566667\pi$
$461$		0			0		0.207912	+	0.978148i	$0.433333\pi$
$462$		0			0
$463$	0.170401	+	0.262394i	0.170401	+	0.262394i	0.913545	−	0.406737i	$-0.133333\pi$
$463$	0.170401	+	0.262394i	0.170401	+	0.262394i	−0.743145	+	0.669131i	$0.766667\pi$
$464$		0			0
$465$	0.278240	−	0.309017i	0.278240	−	0.309017i
$466$		0			0
$467$	−1.90807	−	0.302208i	−1.90807	−	0.302208i	−0.913545	−	0.406737i	$-0.866667\pi$
$467$	−1.90807	−	0.302208i	−1.90807	−	0.302208i	−0.994522	+	0.104528i	$0.966667\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	1.56593	+	1.01693i	1.56593	+	1.01693i
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−0.0570084	−	1.08779i	−0.0570084	−	1.08779i
$478$		0			0
$479$		0			0		0.104528	−	0.994522i	$-0.466667\pi$
$479$		0			0		−0.104528	+	0.994522i	$0.533333\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	−0.743145	−	0.669131i	−0.743145	−	0.669131i
$485$	1.30354	−	1.05558i	1.30354	−	1.05558i
$486$		0			0
$487$	−0.461219	−	0.235003i	−0.461219	−	0.235003i	0.207912	−	0.978148i	$-0.433333\pi$
$487$	−0.461219	−	0.235003i	−0.461219	−	0.235003i	−0.669131	+	0.743145i	$0.733333\pi$
$488$		0			0
$489$	−0.0270911	−	0.101105i	−0.0270911	−	0.101105i
$490$		0			0
$491$		0			0		0.207912	−	0.978148i	$-0.433333\pi$
$491$		0			0		−0.207912	+	0.978148i	$0.566667\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	0.104528	+	0.994522i	0.104528	+	0.994522i
$496$	−0.336408	−	0.244415i	−0.336408	−	0.244415i
$497$		0			0
$498$		0			0
$499$	−1.28716	−	0.743145i	−1.28716	−	0.743145i	−0.309017	−	0.951057i	$-0.600000\pi$
$499$	−1.28716	−	0.743145i	−1.28716	−	0.743145i	−0.978148	+	0.207912i	$0.933333\pi$
$500$	−0.913545	+	0.406737i	−0.913545	+	0.406737i
$501$		0			0
$502$		0			0
$503$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$503$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	0.309017	+	0.951057i	0.309017	+	0.951057i
$508$		0			0
$509$	1.58268	+	0.336408i	1.58268	+	0.336408i	0.913545	−	0.406737i	$-0.133333\pi$
$509$	1.58268	+	0.336408i	1.58268	+	0.336408i	0.669131	+	0.743145i	$0.266667\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	0.557008	−	1.45106i	0.557008	−	1.45106i
$516$		0			0
$517$	−0.402280	−	0.325760i	−0.402280	−	0.325760i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	1.07394	−	1.47815i	1.07394	−	1.47815i	0.207912	−	0.978148i	$-0.433333\pi$
$521$	1.07394	−	1.47815i	1.07394	−	1.47815i	0.866025	−	0.500000i	$-0.166667\pi$
$522$		0			0
$523$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$523$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$	0.978148	−	0.207912i	0.978148	−	0.207912i
$529$	2.88621	+	0.303353i	2.88621	+	0.303353i
$530$		0			0
$531$	−0.413545	+	1.27276i	−0.413545	+	1.27276i
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−1.97815	+	0.207912i	−1.97815	+	0.207912i
$538$		0			0
$539$	0.309017	−	0.951057i	0.309017	−	0.951057i
$540$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$541$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$541$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$542$		0			0
$543$	−0.360114	+	1.69420i	−0.360114	+	1.69420i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		−0.629320	−	0.777146i	$-0.716667\pi$
$547$		0			0		0.629320	+	0.777146i	$0.283333\pi$
$548$	−0.705634	−	1.38488i	−0.705634	−	1.38488i
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$	0.500000	+	1.86603i	0.500000	+	1.86603i
$556$		0			0
$557$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$557$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.838671	−	0.544639i	$-0.183333\pi$
$563$		0			0		−0.838671	+	0.544639i	$0.816667\pi$
$564$	0.500000	−	0.133975i	0.500000	−	0.133975i
$565$	−1.01693	−	1.56593i	−1.01693	−	1.56593i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		−0.913545	−	0.406737i	$-0.866667\pi$
$569$		0			0		0.913545	+	0.406737i	$0.133333\pi$
$570$		0			0
$571$		0			0		0.913545	−	0.406737i	$-0.133333\pi$
$571$		0			0		−0.913545	+	0.406737i	$0.866667\pi$
$572$		0			0
$573$	0.209057			0.209057
$574$		0			0
$575$	−1.97267	−	0.103383i	−1.97267	−	0.103383i
$576$	−0.406737	+	0.913545i	−0.406737	+	0.913545i
$577$	0.906737	+	1.77957i	0.906737	+	1.77957i	0.500000	+	0.866025i	$0.333333\pi$
$577$	0.906737	+	1.77957i	0.906737	+	1.77957i	0.406737	+	0.913545i	$0.366667\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−0.390362	−	1.01693i	−0.390362	−	1.01693i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	0.104528	−	0.00547810i	0.104528	−	0.00547810i		−	1.00000i	$-0.5\pi$
$587$	0.104528	−	0.00547810i	0.104528	−	0.00547810i	0.104528	+	0.994522i	$0.466667\pi$
$588$	0.587785	+	0.809017i	0.587785	+	0.809017i
$589$		0			0
$590$		0			0
$591$		0			0
$592$	1.80354	−	0.692314i	1.80354	−	0.692314i
$593$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$593$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−0.544320	−	0.604528i	−0.544320	−	0.604528i
$598$		0			0
$599$	0.951057	−	1.64728i	0.951057	−	1.64728i	0.207912	−	0.978148i	$-0.433333\pi$
$599$	0.951057	−	1.64728i	0.951057	−	1.64728i	0.743145	−	0.669131i	$-0.233333\pi$
$600$		0			0
$601$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$601$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$602$		0			0
$603$	0.506809	−	0.506809i	0.506809	−	0.506809i
$604$		0			0
$605$	0.406737	+	0.913545i	0.406737	+	0.913545i
$606$		0			0
$607$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$607$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$613$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−1.05558	+	1.30354i	−1.05558	+	1.30354i	−0.104528	+	0.994522i	$0.533333\pi$
$617$	−1.05558	+	1.30354i	−1.05558	+	1.30354i	−0.951057	+	0.309017i	$0.900000\pi$
$618$		0			0
$619$	−0.795697	+	1.78716i	−0.795697	+	1.78716i	−0.207912	+	0.978148i	$0.566667\pi$
$619$	−0.795697	+	1.78716i	−0.795697	+	1.78716i	−0.587785	+	0.809017i	$0.700000\pi$
$620$	0.207912	+	0.360114i	0.207912	+	0.360114i
$621$	−1.07587	−	1.65669i	−1.07587	−	1.65669i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	1.00000			1.00000
$626$		0			0
$627$		0			0
$628$	−1.45106	+	1.17504i	−1.45106	+	1.17504i
$629$		0			0
$630$		0			0
$631$	1.47815	−	1.07394i	1.47815	−	1.07394i	0.500000	−	0.866025i	$-0.333333\pi$
$631$	1.47815	−	1.07394i	1.47815	−	1.07394i	0.978148	−	0.207912i	$-0.0666667\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$	1.05216	+	0.281926i	1.05216	+	0.281926i
$637$		0			0
$638$		0			0
$639$	1.10453	+	0.994522i	1.10453	+	0.994522i
$640$		0			0
$641$	−0.459289	−	0.413545i	−0.459289	−	0.413545i	0.406737	−	0.913545i	$-0.366667\pi$
$641$	−0.459289	−	0.413545i	−0.459289	−	0.413545i	−0.866025	+	0.500000i	$0.833333\pi$
$642$		0			0
$643$	−0.877042	+	0.235003i	−0.877042	+	0.235003i	−0.669131	−	0.743145i	$-0.733333\pi$
$643$	−0.877042	+	0.235003i	−0.877042	+	0.235003i	−0.207912	+	0.978148i	$0.566667\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−0.0489435	−	0.309017i	−0.0489435	−	0.309017i	0.951057	−	0.309017i	$-0.100000\pi$
$647$	−0.0489435	−	0.309017i	−0.0489435	−	0.309017i	−1.00000			$\pi$
$648$		0			0
$649$			1.33826i			1.33826i
$650$		0			0
$651$		0			0
$652$	0.104528	+	0.00547810i	0.104528	+	0.00547810i
$653$	0.601105	−	1.56593i	0.601105	−	1.56593i	−0.207912	−	0.978148i	$-0.566667\pi$
$653$	0.601105	−	1.56593i	0.601105	−	1.56593i	0.809017	−	0.587785i	$-0.200000\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		0.669131	−	0.743145i	$-0.266667\pi$
$659$		0			0		−0.669131	+	0.743145i	$0.733333\pi$
$660$	−0.978148	−	0.207912i	−0.978148	−	0.207912i
$661$	−0.139886	−	0.155360i	−0.139886	−	0.155360i	0.669131	−	0.743145i	$-0.266667\pi$
$661$	−0.139886	−	0.155360i	−0.139886	−	0.155360i	−0.809017	+	0.587785i	$0.800000\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−1.76007	+	0.278768i	−1.76007	+	0.278768i
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		−0.838671	−	0.544639i	$-0.816667\pi$
$673$		0			0		0.838671	+	0.544639i	$0.183333\pi$
$674$		0			0
$675$	0.587785	+	0.809017i	0.587785	+	0.809017i
$676$	−1.00000			−1.00000
$677$		0			0		−0.838671	−	0.544639i	$-0.816667\pi$
$677$		0			0		0.838671	+	0.544639i	$0.183333\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	0.196895	−	1.24314i	0.196895	−	1.24314i	−0.669131	−	0.743145i	$-0.733333\pi$
$683$	0.196895	−	1.24314i	0.196895	−	1.24314i	0.866025	−	0.500000i	$-0.166667\pi$
$684$		0			0
$685$	0.0813454	+	1.55216i	0.0813454	+	1.55216i
$686$		0			0
$687$	1.07394	+	0.478148i	1.07394	+	0.478148i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−0.139886	+	0.155360i	−0.139886	+	0.155360i	−0.809017	−	0.587785i	$-0.800000\pi$
$691$	−0.139886	+	0.155360i	−0.139886	+	0.155360i	0.669131	+	0.743145i	$0.266667\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$		0			0
$703$		0			0
$704$	−0.104528	+	0.994522i	−0.104528	+	0.994522i
$705$	−0.511265	−	0.0809764i	−0.511265	−	0.0809764i
$706$		0			0
$707$		0			0
$708$	−1.08268	−	0.786610i	−1.08268	−	0.786610i
$709$	0.994522	+	0.895472i	0.994522	+	0.895472i	0.994522	−	0.104528i	$-0.0333333\pi$
$709$	0.994522	+	0.895472i	0.994522	+	0.895472i			1.00000i	$0.5\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	0.0429892	+	0.820282i	0.0429892	+	0.820282i
$714$		0			0
$715$		0			0
$716$	0.413545	−	1.94558i	0.413545	−	1.94558i
$717$		0			0
$718$		0			0
$719$	0.658114	−	0.478148i	0.658114	−	0.478148i	−0.207912	−	0.978148i	$-0.566667\pi$
$719$	0.658114	−	0.478148i	0.658114	−	0.478148i	0.866025	+	0.500000i	$0.166667\pi$
$720$	0.743145	−	0.669131i	0.743145	−	0.669131i
$721$		0			0
$722$		0			0
$723$		0			0
$724$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$725$		0			0
$726$		0			0
$727$	0.593263	−	0.913545i	0.593263	−	0.913545i	−0.406737	−	0.913545i	$-0.633333\pi$
$727$	0.593263	−	0.913545i	0.593263	−	0.913545i	1.00000			$0$
$728$		0			0
$729$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		0.629320	−	0.777146i	$-0.283333\pi$
$733$		0			0		−0.629320	+	0.777146i	$0.716667\pi$
$734$		0			0
$735$	−0.207912	−	0.978148i	−0.207912	−	0.978148i
$736$		0			0
$737$	0.325391	−	0.638616i	0.325391	−	0.638616i
$738$		0			0
$739$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$739$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$740$	−1.92920	−	0.101105i	−1.92920	−	0.101105i
$741$		0			0
$742$		0			0
$743$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$743$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	0.913545	−	1.58231i	0.913545	−	1.58231i	0.104528	−	0.994522i	$-0.466667\pi$
$751$	0.913545	−	1.58231i	0.913545	−	1.58231i	0.809017	−	0.587785i	$-0.200000\pi$
$752$	−0.0270911	+	0.516929i	−0.0270911	+	0.516929i
$753$	1.27276	+	1.41355i	1.27276	+	1.41355i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	1.41228	−	1.41228i	1.41228	−	1.41228i	0.669131	−	0.743145i	$-0.266667\pi$
$757$	1.41228	−	1.41228i	1.41228	−	1.41228i	0.743145	−	0.669131i	$-0.233333\pi$
$758$		0			0
$759$	−1.53516	−	1.24314i	−1.53516	−	1.24314i
$760$		0			0
$761$		0			0		0.743145	−	0.669131i	$-0.233333\pi$
$761$		0			0		−0.743145	+	0.669131i	$0.766667\pi$
$762$		0			0
$763$		0			0
$764$	−0.0646021	+	0.198825i	−0.0646021	+	0.198825i
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−0.743145	−	0.669131i	−0.743145	−	0.669131i
$769$		0			0		0.994522	−	0.104528i	$-0.0333333\pi$
$769$		0			0		−0.994522	+	0.104528i	$0.966667\pi$
$770$		0			0
$771$	0.170401	−	0.262394i	0.170401	−	0.262394i
$772$		0			0
$773$	−0.809017	−	1.58779i	−0.809017	−	1.58779i	−0.809017	−	0.587785i	$-0.800000\pi$
$773$	−0.809017	−	1.58779i	−0.809017	−	1.58779i		−	1.00000i	$-0.5\pi$
$774$		0			0
$775$	−0.0434654	−	0.413545i	−0.0434654	−	0.413545i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	1.35779	+	0.604528i	1.35779	+	0.604528i
$782$		0			0
$783$		0			0
$784$	−0.951057	+	0.309017i	−0.951057	+	0.309017i
$785$	1.80354	−	0.483257i	1.80354	−	0.483257i
$786$		0			0
$787$		0			0		0.838671	−	0.544639i	$-0.183333\pi$
$787$		0			0		−0.838671	+	0.544639i	$0.816667\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$	−0.846528	−	0.685505i	−0.846528	−	0.685505i
$796$	0.743145	−	0.330869i	0.743145	−	0.330869i
$797$	−1.17504	−	0.451057i	−1.17504	−	0.451057i	−0.309017	−	0.951057i	$-0.600000\pi$
$797$	−1.17504	−	0.451057i	−1.17504	−	0.451057i	−0.866025	+	0.500000i	$0.833333\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	1.07394	+	0.478148i	1.07394	+	0.478148i
$802$		0			0
$803$		0			0
$804$	0.325391	+	0.638616i	0.325391	+	0.638616i
$805$		0			0
$806$		0			0
$807$	0.379874	−	1.78716i	0.379874	−	1.78716i
$808$		0			0
$809$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$809$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$810$		0			0
$811$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$811$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−0.0932634	−	0.0475201i	−0.0932634	−	0.0475201i
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		−0.994522	−	0.104528i	$-0.966667\pi$
$821$		0			0		0.994522	+	0.104528i	$0.0333333\pi$
$822$		0			0
$823$	0.0932634	−	1.77957i	0.0932634	−	1.77957i	−0.406737	−	0.913545i	$-0.633333\pi$
$823$	0.0932634	−	1.77957i	0.0932634	−	1.77957i	0.500000	−	0.866025i	$-0.333333\pi$
$824$		0			0
$825$	0.809017	+	0.587785i	0.809017	+	0.587785i
$826$		0			0
$827$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$827$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$828$	1.90807	−	0.511265i	1.90807	−	0.511265i
$829$	1.14988	−	1.58268i	1.14988	−	1.58268i	0.406737	−	0.913545i	$-0.366667\pi$
$829$	1.14988	−	1.58268i	1.14988	−	1.58268i	0.743145	−	0.669131i	$-0.233333\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	0.309017	−	0.278240i	0.309017	−	0.278240i
$838$		0			0
$839$	−1.58268	+	0.336408i	−1.58268	+	0.336408i	−0.913545	−	0.406737i	$-0.866667\pi$
$839$	−1.58268	+	0.336408i	−1.58268	+	0.336408i	−0.669131	+	0.743145i	$0.733333\pi$
$840$		0			0
$841$	−0.978148	−	0.207912i	−0.978148	−	0.207912i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	0.913545	+	0.406737i	0.913545	+	0.406737i
$846$		0			0
$847$		0			0
$848$	−0.593263	+	0.913545i	−0.593263	+	0.913545i
$849$		0			0
$850$		0			0
$851$	−3.30487	−	1.90807i	−3.30487	−	1.90807i
$852$	−1.28716	+	0.743145i	−1.28716	+	0.743145i
$853$		0			0		0.777146	−	0.629320i	$-0.216667\pi$
$853$		0			0		−0.777146	+	0.629320i	$0.783333\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$857$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$858$		0			0
$859$	0.406737	−	1.91355i	0.406737	−	1.91355i		−	1.00000i	$-0.5\pi$
$859$	0.406737	−	1.91355i	0.406737	−	1.91355i	0.406737	−	0.913545i	$-0.366667\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−1.58779	−	0.809017i	−1.58779	−	0.809017i	−0.587785	−	0.809017i	$-0.700000\pi$
$863$	−1.58779	−	0.809017i	−1.58779	−	0.809017i	−1.00000			$\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	0.913545	−	0.406737i	0.913545	−	0.406737i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	1.40674	−	0.913545i	1.40674	−	0.913545i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		−0.998630	−	0.0523360i	$-0.983333\pi$
$877$		0			0		0.998630	+	0.0523360i	$0.0166667\pi$
$878$		0			0
$879$		0			0
$880$	0.500000	−	0.866025i	0.500000	−	0.866025i
$881$	0.244415	+	0.336408i	0.244415	+	0.336408i	0.913545	−	0.406737i	$-0.133333\pi$
$881$	0.244415	+	0.336408i	0.244415	+	0.336408i	−0.669131	+	0.743145i	$0.733333\pi$
$882$		0			0
$883$	1.97267	+	0.312440i	1.97267	+	0.312440i	0.994522	+	0.104528i	$0.0333333\pi$
$883$	1.97267	+	0.312440i	1.97267	+	0.312440i	0.978148	+	0.207912i	$0.0666667\pi$
$884$		0			0
$885$	0.669131	+	1.15897i	0.669131	+	1.15897i
$886$		0			0
$887$		0			0		−0.544639	−	0.838671i	$-0.683333\pi$
$887$		0			0		0.544639	+	0.838671i	$0.316667\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$			1.00000i			1.00000i
$892$	0.278768	−	1.76007i	0.278768	−	1.76007i
$893$		0			0
$894$		0			0
$895$	−1.16913	+	1.60917i	−1.16913	+	1.60917i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−0.951057	+	0.309017i	−0.951057	+	0.309017i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	1.01807	+	1.40126i	1.01807	+	1.40126i
$906$		0			0
$907$	−1.36603	−	0.366025i	−1.36603	−	0.366025i	−0.500000	−	0.866025i	$-0.666667\pi$
$907$	−1.36603	−	0.366025i	−1.36603	−	0.366025i	−0.866025	+	0.500000i	$0.833333\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	0.0434654	+	0.204489i	0.0434654	+	0.204489i	0.994522	−	0.104528i	$-0.0333333\pi$
$911$	0.0434654	+	0.204489i	0.0434654	+	0.204489i	−0.951057	+	0.309017i	$0.900000\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−0.786610	+	0.873619i	−0.786610	+	0.873619i
$917$		0			0
$918$		0			0
$919$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$919$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	1.72129	+	0.877042i	1.72129	+	0.877042i
$926$		0			0
$927$	0.705634	−	1.38488i	0.705634	−	1.38488i
$928$		0			0
$929$	0.0218524	−	0.207912i	0.0218524	−	0.207912i	−0.978148	−	0.207912i	$-0.933333\pi$
$929$	0.0218524	−	0.207912i	0.0218524	−	0.207912i	1.00000			$0$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	0.0850311	−	0.809017i	0.0850311	−	0.809017i
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$937$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$938$		0			0
$939$	0.221232	−	0.221232i	0.221232	−	0.221232i
$940$	0.235003	−	0.461219i	0.235003	−	0.461219i
$941$		0			0		0.207912	−	0.978148i	$-0.433333\pi$
$941$		0			0		−0.207912	+	0.978148i	$0.566667\pi$
$942$		0			0
$943$		0			0
$944$	1.08268	−	0.786610i	1.08268	−	0.786610i
$945$		0			0
$946$		0			0
$947$	0.0813454	−	0.0658722i	0.0813454	−	0.0658722i	−0.587785	−	0.809017i	$-0.700000\pi$
$947$	0.0813454	−	0.0658722i	0.0813454	−	0.0658722i	0.669131	+	0.743145i	$0.266667\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$	−0.809017	−	0.412215i	−0.809017	−	0.412215i
$952$		0			0
$953$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$953$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$954$		0			0
$955$	0.139886	−	0.155360i	0.139886	−	0.155360i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$	0.406737	+	0.913545i	0.406737	+	0.913545i
$961$	0.809017	−	0.171962i	0.809017	−	0.171962i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0		−0.777146	−	0.629320i	$-0.783333\pi$
$967$		0			0		0.777146	+	0.629320i	$0.216667\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	0.363271	−	0.500000i	0.363271	−	0.500000i	−0.587785	−	0.809017i	$-0.700000\pi$
$971$	0.363271	−	0.500000i	0.363271	−	0.500000i	0.951057	+	0.309017i	$0.100000\pi$
$972$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	0.0740142	−	1.41228i	0.0740142	−	1.41228i	−0.669131	−	0.743145i	$-0.733333\pi$
$977$	0.0740142	−	1.41228i	0.0740142	−	1.41228i	0.743145	−	0.669131i	$-0.233333\pi$
$978$		0			0
$979$	1.16913	+	0.122881i	1.16913	+	0.122881i
$980$	0.994522	+	0.104528i	0.994522	+	0.104528i
$981$		0			0
$982$		0			0
$983$	1.17504	−	0.451057i	1.17504	−	0.451057i	0.309017	−	0.951057i	$-0.400000\pi$
$983$	1.17504	−	0.451057i	1.17504	−	0.451057i	0.866025	+	0.500000i	$0.166667\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−0.587785	−	1.80902i	−0.587785	−	1.80902i	−0.587785	−	0.809017i	$-0.700000\pi$
$991$	−0.587785	−	1.80902i	−0.587785	−	1.80902i		−	1.00000i	$-0.5\pi$
$992$		0			0
$993$	−1.89169	+	0.614648i	−1.89169	+	0.614648i
$994$		0			0
$995$	−0.813473			−0.813473
$996$		0			0
$997$		0			0		−0.629320	−	0.777146i	$-0.716667\pi$
$997$		0			0		0.629320	+	0.777146i	$0.283333\pi$
$998$		0			0
$999$	0.302208	+	1.90807i	0.302208	+	1.90807i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2475.1.fy.b.428.1	yes	16
9.2	odd	6		2475.1.fy.a.1253.1	✓	16
11.10	odd	2	CM	2475.1.fy.b.428.1	yes	16
25.17	odd	20		2475.1.fy.a.1517.1	yes	16
99.65	even	6		2475.1.fy.a.1253.1	✓	16
225.92	even	60	inner	2475.1.fy.b.2342.1	yes	16
275.142	even	20		2475.1.fy.a.1517.1	yes	16
2475.2342	odd	60	inner	2475.1.fy.b.2342.1	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2475.1.fy.a.1253.1	✓	16	9.2	odd	6
2475.1.fy.a.1253.1	✓	16	99.65	even	6
2475.1.fy.a.1517.1	yes	16	25.17	odd	20
2475.1.fy.a.1517.1	yes	16	275.142	even	20
2475.1.fy.b.428.1	yes	16	1.1	even	1	trivial
2475.1.fy.b.428.1	yes	16	11.10	odd	2	CM
2475.1.fy.b.2342.1	yes	16	225.92	even	60	inner
2475.1.fy.b.2342.1	yes	16	2475.2342	odd	60	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2475.fy (of order \(60\), degree \(16\), minimal)

\(f(q)\)	\(=\)	\(q+(0.743145 - 0.669131i) q^{3} +(0.406737 + 0.913545i) q^{4} -1.00000i q^{5} +(0.104528 - 0.994522i) q^{9} +O(q^{10})\)	\(q+(0.743145 - 0.669131i) q^{3} +(0.406737 + 0.913545i) q^{4} -1.00000i q^{5} +(0.104528 - 0.994522i) q^{9} +(-0.207912 - 0.978148i) q^{11} +(0.913545 + 0.406737i) q^{12} +(-0.669131 - 0.743145i) q^{15} +(-0.669131 + 0.743145i) q^{16} +(0.913545 - 0.406737i) q^{20} +(1.97267 + 0.103383i) q^{23} -1.00000 q^{25} +(-0.587785 - 0.809017i) q^{27} +(0.0434654 + 0.413545i) q^{31} +(-0.809017 - 0.587785i) q^{33} +(0.951057 - 0.309017i) q^{36} +(-1.72129 - 0.877042i) q^{37} +(0.809017 - 0.587785i) q^{44} +(-0.994522 - 0.104528i) q^{45} +(0.402280 - 0.325760i) q^{47} +1.00000i q^{48} +(0.866025 + 0.500000i) q^{49} +(1.07587 - 0.170401i) q^{53} +(-0.978148 + 0.207912i) q^{55} +(-1.30902 - 0.278240i) q^{59} +(0.406737 - 0.913545i) q^{60} +(-0.951057 - 0.309017i) q^{64} +(0.557008 + 0.451057i) q^{67} +(1.53516 - 1.24314i) q^{69} +(-0.873619 + 1.20243i) q^{71} +(-0.743145 + 0.669131i) q^{75} +(0.743145 + 0.669131i) q^{80} +(-0.978148 - 0.207912i) q^{81} +(-0.363271 + 1.11803i) q^{89} +(0.707912 + 1.84417i) q^{92} +(0.309017 + 0.278240i) q^{93} +(1.05558 + 1.30354i) q^{97} +(-0.994522 + 0.104528i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 2 q^{9}+O(q^{10}) \) 16 * q - 2 * q^9	\( 16 q - 2 q^{9} + 2 q^{12} - 2 q^{15} - 2 q^{16} + 2 q^{20} - 2 q^{23} - 16 q^{25} - 4 q^{33} + 2 q^{37} + 4 q^{44} + 4 q^{47} + 2 q^{53} + 2 q^{55} - 12 q^{59} + 4 q^{67} + 2 q^{69} + 2 q^{81} + 8 q^{92} - 4 q^{93} - 2 q^{97}+O(q^{100}) \) 16 * q - 2 * q^9 + 2 * q^12 - 2 * q^15 - 2 * q^16 + 2 * q^20 - 2 * q^23 - 16 * q^25 - 4 * q^33 + 2 * q^37 + 4 * q^44 + 4 * q^47 + 2 * q^53 + 2 * q^55 - 12 * q^59 + 4 * q^67 + 2 * q^69 + 2 * q^81 + 8 * q^92 - 4 * q^93 - 2 * q^97

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